Abstract
Three cases of weakly nonlinear propagation of plane progressive pressure waves in flowing water containing many uniformly separated spherical microbubbles are theoretically investigated, with a particular focus on different cases of initial flow patterns. The gas and liquid phases initially have a high velocity with uniform distribution, and a low velocity with nonuniform distribution for each phase. Basic conservation laws based on a two-fluid model are employed to govern different velocity distributions for gas and liquid phases. From the method of multiple scales with perturbation expansions, we derive three cases of nonlinear wave equation describing the long-range propagation of waves: (i) the KdVB (Korteweg–de Vries–Burgers) equation for a long wave, (ii) the NLS (nonlinear Schrödinger)-I equation for a short wave in slow nonuniform flow, (iii) the NLS-II equation for a short wave in fast nonuniform flow. As a result, all the initial velocities contribute to an advection effect of waves, initial nonuniform flow distribution induces a variable coefficient of advection term into the KdVB, NLS-I, and NLS-II equations, and an important role of relative velocity between the gas and liquid phases is clarified.
Published Version
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